final_quiz_1.tex

\yourname

\activitytitle{Final quiz}{30 points}

\noindent
Please use one side of a sheet of paper for each problem.
If you are totally stuck, you can ask for a hint, but it may cost you a little something.
Good luck!

\vspace{0.2in}

\blist{0.0in}
\item a) State the definition of odd.

b) Show that the product of two odd numbers is odd.
Do your best to write a picture--perfect proof.

c) Let $P(n)$ be the statement ``The product of $n$ odd numbers is odd.''
You may want to write this as ``$k_1 k_2 \cdots k_n$ is odd.''
Show that $P(n)$ is true for all $n = 1, 2, \ldots$.

\vfill


\item Answer this question on the next page.

a) Let $n$ be an integer.
Explain how you know that you can write $n = 5q + r$ where $q$ and $r$ are integers and $0 \leq r < 5$.

b) Let $m$ be an integer and suppose that $5m = 7n$.
Show that $n$ is a multiple of 5.
Don't use prime factorization.

\pagebreak

\item Let $X$ be the set of all ordered pairs of integers.
Then $X$ contains things like $(1,3), (-4,11),$ and $(9,7)$.
We say that $(a,b) \sim (p,q)$ if $aq = bp$.

\noindent
a) Is $(6,10) \sim (9,15)$?

\vspace*{1in}
\noindent
b) Show that $\sim$ is reflexive.

\vspace*{1in}
\noindent
c) Show that $\sim$ is symmetric.

\vspace*{1in}
\noindent
d) Show that $\sim$ is transitive.

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\noindent
e) Describe all members of the equivalence class containing $(6,10)$.

\vfill

\item Answer this question on the next page.
Show that $\bigcup_{n=1}^{\infty} [2, 5-\frac{1}{n}] = [2,5)$.  Draw pictures, then show set inclusion both ways.

\pagebreak

\definitionNN{176. Positive real numbers}{By construction, the real numbers have a subset $\Rp$, called the {\em positive real numbers,} for which:
\balist{0.1in}
\item If $a,b \in \Rp$, then $a + b \in \Rp$.  ($\Rp$ is closed under addition.)
\item If $a,b \in \Rp$, then $a\cdot b \in \Rp$.  ($\Rp$ is closed under multiplication.)
\item For every real number $a$, either $a \in \Rp$ or $(-a) \in \Rp$ or $a = 0$.  Exactly one of the three happens.
\ealist
}

\item Let $a \in \R$ with $a \ne 0$.
Show that $a\cdot a \in \Rp$.
Be extremely clear about every step that you take.

\vfill

\definitionNN{Absolute value}{The function $f : \R \to \R$ defined by
\[
    f(x) = \left\{ \begin{array}{cl} x, & \mbox{if $x \geq 0$} \\
                          -x, & \mbox{if $x < 0$} \end{array} \right.
\]
is called the {\em absolute value} function.}

\item Answer this question on the next page.
Show that $f(-a) = f(a)$ for all real numbers $a$. 

\elist